Canonical quantization of non-commutative holonomies in 2+1 loop quantum gravity

TL;DR

This paper develops a canonical quantization of non-commutative holonomies in 2+1 loop quantum gravity with a positive cosmological constant, revealing a connection to Kauffman's q-deformed crossing identity within standard SU(2) spin networks.

Contribution

It introduces a quantum holonomy operator acting on SU(2) spin networks, linking its crossing behavior to Kauffman's q-deformation without q-deformed spin networks.

Findings

Quantum holonomy acts non-trivially at crossings.

Connection to Kauffman's q-deformed crossing identity.

Operates within standard SU(2) spin network framework.

Abstract

In this work we investigate the canonical quantization of 2+1 gravity with cosmological constant $\Lambda>0$ in the canonical framework of loop quantum gravity. The unconstrained phase space of gravity in 2+1 dimensions is coordinatized by an SU(2) connection $A$ and the canonically conjugate triad field $e$. A natural regularization of the constraints of 2+1 gravity can be defined in terms of the holonomies of $A+=A + \sqrt\Lambda e$. As a first step towards the quantization of these constraints we study the canonical quantization of the holonomy of the connection $A_{\lambda}=A+\lambda e$ on the kinematical Hilbert space of loop quantum gravity. The holonomy operator associated to a given path acts non trivially on spin network links that are transversal to the path (a crossing). We provide an explicit construction of the quantum holonomy operator. In particular, we exhibit a close relationship between the action of the quantum holonomy at a crossing and Kauffman's q-deformed crossing identity. The crucial difference is that (being an operator acting on the kinematical Hilbert space of LQG) the result is completely described in terms of standard SU(2) spin network states (in contrast to q-deformed spin networks in Kauffman's identity). We discuss the possible implications of our result.